Methods and systems for power systems analysis

ABSTRACT

Methods, systems and software products are provided for power system analysis, including a synchronized-measurement-based (also referred to as Phasor Measurement-based) one-shot non-iterative solution to the well-known power flow (or load flow) problem. A set of non-linear equations, quadratic or third-order in form and containing power system parameters and unknowns, are formed and subsequently transformed into a set of linear equations. Subsequently a selected set of variables (or unknowns) are solved for in terms of the remaining variables. A second set of quadratic equations is then defined and used to solve for the selected variables, resulting in a solution set for the non-linear equations. The methodology may be iterative or non-iterative, and may be used for the solution of a number of power system analysis problems to include: AC power flow, AC state estimation, optimal power flow, stability analysis economic dispatch, and unit commitment.

RELATED APPLICATIONS

This non-provisional application claims priority to provisional application No. 60/536,660, filed on Jan. 15, 2004 and incorporated herein by reference.

BACKGROUND

Electric power systems are comprised of generating stations or power plants, transmission and distribution lines interconnected into networks, a variety of loads, and many other necessary pieces of equipment. Electrical power systems' operation and control functions rely on analytical tools for setting safe and secure operating limits and maintaining reliable service. These tools utilize mathematical models of the power system components and equipment in the form of differential and algebraic equations. These mathematical models are formulated and coded in the form of computer programs used to determine the power system operating state or its response to various changes and disturbances such as equipment switchings as well as faults. The formulated problems are based on ohm's and kirchhoff's laws. The solutions to many of these formulated analysis problems are heavily relied upon for monitoring and managing the performance of power systems.

Power systems are typically analyzed in both an “off-line” and an “on-line” (near real-time) manner. Off-line analysis tools utilize alternating current (AC) power flow and stability solutions for the purpose of system planning, expansion, and design. Stability algorithms or programs simulate the dynamics of the power system, and require that, for each integration step of the differential equations being solved, an AC power flow solution be performed.

On-line analysis tools, on the other hand, are utilized for near real-time operation and control of power systems at utility or Independent System Operator (ISO) or Regional Transmission Operator (RTO) energy control centers. These analysis tools typically require telemetered data collected via “supervisory control and data acquisition” (SCADA) systems. SCADA systems and associated computational tools are collectively referred to as an Energy Management System (EMS). In the on-line analysis, the collected telemetry asynchronous data is applied or provided to a tool referred to as a State Estimator (SE). The SE estimates the state (voltage magnitude and phase-angle at all the buses) of the power system, as well as the resulting power flow values, using conventional best-fit procedures. For example, the Least-Squares best fit is one conventional estimating algorithm that may be used by the SE tool. The SE output is then used to determine system parameters that are not directly measured or measurable. Other functions such as operator's power flow, contingency ranking and analysis, economic dispatch, etc., may also be implemented using the SE output.

AC power flow and AC state estimation algorithms are similar in nature, and generally use a successive, iterative linearized approximation of a set of non-linear equations. Newton and Newton-Raphson methods or variations thereof are conventionally used to iteratively approximate solutions to the set of non-linear equations. Newton-based iterative solution methods, for example, begin with an initial estimate of the solution at a point (in a vector sense) where a matrix of gradients or partial derivatives, referred to as “the Jacobian,” may be determined. This matrix is used to determine a correction to the initial estimated solution using a linear approximation of the non-linear equations. The correction is then added to the initial estimate, which is then used as a new point in the iterative process. Iterations of the solution are continued until the difference between two consecutive solution points is less than a specified tolerance, i.e., a convergence criterion is satisfied or convergence is achieved. Newton-type methods, although more computationally intensive, normally converge in a few iterations; however, these methods require that the initial solution estimate be close to the true or actual solution.

Another conventional AC power flow solution algorithm, referred to as Gauss-Seidel, performs power system analysis by starting from an initial estimate and iterating on a set of the complex form of non-linear power flow equations. This algorithm does not take advantage of gradient or derivative information, however, and generally requires a large number of iterations for convergence. The algorithm is therefore generally employed when the initial guess may be further from the true or actual solution and when the solution speed is of no concern.

The prior art discussed above thus depends on the quality of the initial estimate. Moreover, the dynamic fluctuations in the operating state of power systems limit the use of iterative solutions in the real-time determination of the state of the power system for automated operation and control.

SUMMARY

In particular, and by way of example only, according to an embodiment, provided is a method for analyzing a power system, comprising: formulating a set of non-linear equations representative of a power system and containing parameters and unknowns; transforming the set of non-linear equations into a corresponding first set of quadratic equations; subsequently changing variables of the first set of quadratic equations to define a second set of quadratic equations; and solving the second set of quadratic equations to define a solution set for the power system unknowns.

In one embodiment, provided is a system for analyzing a power system comprising: an algorithm for transforming a set of non-linear equations, representative of a power system and having parameters and unknowns, into a first set of linear equations having a coefficients matrix, a variables vector and a right hand side vector; an algorithm for solving for a first variable set in terms of a second variable set in the first set of linear equations, for formulating a second set of quadratic equations, and for transforming the second set of quadratic equations into a second set of linear equations wherein a third variable set is defined; an algorithm for calculating the first, second and third variable sets, and for calculating a solution set for the set of non-linear equations; and a processor for the executing, the transforming, the solving and the calculating algorithms, and for applying the solution set to the set of non-linear equations to solve for the unknowns.

In yet another embodiment, provided is a method for analyzing a power system comprising: formulating a set of nonlinear equations representative of a power system and having parameters and unknowns; transforming the set of non-linear equations into a corresponding first set of quadratic equations; expressing the first set of quadratic equations as a first set of linear equations having a coefficients matrix, a variables vector and a right hand side vector; defining a first variable set in the first set of linear equations in terms of a second variable set; formulating a second set of quadratic equations in terms of the second variable set; converting the second set of quadratic equations into a second set of linear equations defined in terms of a third variable set; solving for the third variable set; and calculating the first variable set, the second variable set and the third variable set to define a solution set for the power system unknowns.

In one embodiment, provided is a method for analyzing a power system, comprising: defining a set of non-linear equations representative of a power system and containing power system parameters and unknowns; inputting power system parameters and inputting power system measurements; selecting between an iterative and a non-iterative analytical approach to calculate a solution set for the set of non-linear equations, wherein the non-iterative analytical approach is selected if an adequate set of synchronous power system measurements are available; otherwise estimating the required number of power system unknowns; calculating the solution set for the set of non-linear equations, to solve for the remaining power system unknowns; and iterating the calculation of the solution set, when the iterative analytical approach has been selected, to converge on a solution set within a predetermined tolerance.

In one embodiment, a software product has instructions, stored on computer-readable media, wherein the instructions, when executed by a computer, perform steps for power systems analysis, comprising: instructions for formulating a set of non-linear equations representative of a power system and containing parameters and unknowns; instructions for transforming the set of non-linear equations into a corresponding first set of quadratic equations; instructions for subsequently changing variables of the first set of quadratic equations to define a second set of quadratic equations; and instructions for solving the second set of quadratic equations to define a solution set for the power system unknowns.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow chart illustrating one process embodiment for performing power system analysis.

FIG. 2 is a flow chart illustrating one process embodiment for performing power system analysis.

FIG. 3 is a schematic of one AC power system embodiment.

FIG. 4 is a schematic of one system for executing the processes of FIGS. 1 and 2, in accord with an embodiment.

FIG. 5 shows a process for non-iterative or iterative non-gradient power system analysis, in accord with one embodiment.

DETAILED DESCRIPTION

Following the northeast blackout of Aug. 14, 2003, and based on the recommendations of a joint US/Canadian investigation report, the U.S. Department of Energy initiated the Eastern Interconnection Phasor Project (EIPP). The goal of the project is to instrument the entire eastern interconnection with GPS synchronized measurement systems, also called phasor measurement units (PMU). Working together with North American Electric Reliability Coordinator (NERC), ISOs and many utilities, the project is making real-time synchronously measured data available at central locations via communication networks. Systems, methods and software products disclosed herein provide for utilizing synchronously measured data for a non-iterative, or as appropriate an iterative, solution of the full AC power flow and state estimation problems.

The disclosure herein thus concerns operation, monitoring, management and control of a power system. For example, certain algorithms disclosed herein make possible a synchronized-measurement-based (or phasor-measurement-based) one-shot, non-iterative solution to the well-known power flow (or load flow) and state estimation problems. The disclosed methodologies are inherently parallel. It is anticipated that the combination of the one-shot, non-iterative solution techniques disclosed, and the use of computers with multiple processors, can significantly reduce computation time and improve performance. Reduced computation time allows for real-time analysis and automated control and operation of power systems. See, B. Fardanesh, “Future Trends in Power Systems Control”, IEEE Computer Applications in Power (CAP), Vol. 15, No. 3, July 2002, incorporated herein by reference.

Before proceeding with the detailed description, it is to be appreciated that the present teaching is by way of example, not by limitation. The concepts herein are not limited to use or application with one specific type of analysis of power systems. Thus, although the instrumentalities described herein are for the convenience of explanation, shown and described with respect to exemplary embodiments, it will be appreciated that the principals herein may be equally applied in other types of power system analyses.

In accord with one embodiment, a method solves both an AC power flow and an AC state estimation problem, both of which may be mathematically represented as a set of non-linear equations. The method uses either a direct, non-iterative computational technique or an iterative computational technique, as will be more fully explained below. In the direct, non-iterative solution of the set of nonlinear equations, a direct synchronous measurement of a sufficient number of the system unknowns is available via one or more communication channels. The communication channels may include leased lines, fiber optics, modems, computer networks, etc.

Alternatively, when a sufficient number of measured variable values are not available, an iterative approach requires initial estimates of one or more unknowns so that an interim solution may be found. After making appropriate corrections to the estimates, the solution process repeats and new values for the remaining unknowns are obtained. This iterative process continues until convergence to the true solution is achieved. Equations for solving the Optimal Power Flow (OPF), Economic Dispatch (ED), stability analysis, and Security Constrained Unit Commitment (SCUC) problems can also be solved through a full or partial use of the methodologies disclosed herein.

Non-Iterative Solution Method

FIG. 1 illustrates a flow chart of an exemplary process 100 for a non-iterative solution of an AC power flow or AC state estimation problem, in accordance with an embodiment. In this exemplary process 100, system parameters, which may include bus voltage magnitudes and/or phase angles (phasors), system topology data, system transmission line data, generation and load values, are available and provided (block 105). Phasors are known in the electrical arts as being vector representations of a special type of periodic functions that satisfy a differential equation in !he form of: ${{\frac{\mathbb{d}^{2}}{\mathbb{d}t^{2}}{f(t)}} + {\omega^{2}{f(t)}}} = 0$ where: ω is a constant representing the angular frequency, and the solution is in the form f(t)=A cos ω t+B sin ω t.

Once system parameters have been received and input, non-linear power flow equations are transformed into a set of quadratic equations, block 110. More specifically, using a first change of variables, e.g. x_(i)x_(j)=y_(ij) or x_(i)x_(j)=z_(ij), the quadratic equations are transformed into a set of linear equations consisting of a matrix A, a variables vector VAR, and a right hand side (RHS) vector, in the form of [A][VAR]=[RHS]  (1)

As shown in block 115, matrix A is parsed into a non-singular square matrix portion and the remaining square or rectangular matrix in the form of $\begin{matrix} {{\left\lbrack {MAT} \middle| {ZMAT} \right\rbrack\begin{bmatrix} Y \\ Z \end{bmatrix}} = \lbrack{RHS}\rbrack} & (2) \end{matrix}$ where: [MAT] is a non-singular square matrix;

-   -   [ZMAT] is the remaining square or rectangular matrix of [A]; [Y]         includes the y_(ij) variables associated with matrix [MAT]; and     -   [Z] includes the y_(ij) variables associated with the remaining         square or rectangular matrix, renamed z₁, z₂, . . . z_(k) in the         order they appear.

One skilled in the art has the necessary knowledge to exchange the columns and/or rows of the matrix to ensure the non-singularity of the square partition MAT. The variables vector VAR is also partitioned into a set of “y” and “z” variables (block 120), i.e., those variables corresponding to the square portion and those corresponding to the remaining matrix, respectively. The RHS vector is also partitioned accordingly.

Further, the “y” variables may be determined in terms of the “z” variables in the form of: Y=MAT ⁻¹(RHS−ZMAT·Z)   (3) where each y_(ij) is expressed as a linear combination of the “z” variables, block 125.

Once the variables are defined in terms of “z”, unique variable pair combinations (two-tuple product pairs) are formed providing the basis of a set of independent linear equations, as shown in block 130. In one embodiment, the subroutine for generating all possible unique identifier pairs includes the following steps: (1) get an array Bin from the main program containing the identifiers for they and z variables in the variables vectors, wherein the identifiers are integer numbers with p (an even number) of digits, formed such that Mod(identifier, 10^(p/2)) will result in the index of the one of the original variables; the index for the second variable is found by subtracting 10^(p/2)(Mod(identifier, 10^(p/2))) from the identifier; (2) for all possible pairs of identifiers, pars the two identifiers and obtain four indices; form new identifier pairs by switching the index values in the identifiers, thus resulting in y-variable product pairs in equal quantity; (3) ensure the new equivalent identifier pairs are different than the original pair they resulted from; (4) ensure the new identifiers are legitimate, i.e. contained in the Bin array; and (5) return a vector with all possible unique pairs of identifiers.

As shown in block 135, the product pairs are utilized to generate equations of the following form: y_(ij)y_(mn)=y_(im)y_(jn)= . . . =y_(in)y_(jm)   (4) where: i=0, . . . i max;

-   -   j=0, . . . j max;     -   m=0, . . . m max;     -   n=0, . . . n max: and     -   i max, j max, m max, and n max depend on the total number of         variables and the maximum number of unique two-tuple product         pairs possible.

Still referring to FIG. 1, this new set of quadratic equations is solved using a second change of variables, e.g. z_(i) z_(j)=t, and z_(o) =1, thereby creating a new set of linear equations, block 140. The specific steps of a subroutine used to convert y_(ij)y_(mn)−y_(im)y_(in)=0 quadratic equations in terms of z variables, into a set of linear equations in terms of t variables may include: (1) from the subroutine output get a set of identifiers forming equations of the type y_(ij)y_(mn)−y_(im)y_(jn)=0, wherein each y_(ij) can be given as a vector containing the coefficients of the z terms; (2) form interim matrices by taking the transpose of the coefficient vector corresponding to y_(ij) and multiplying it by the coefficient vector corresponding to y_(mn); (3) form a similar interim matrix for the next product, and subtract the two interim matrices; (4) add the resultant matrix to its transpose. The upper triangular portion of this matrix will contain the coefficients of the set of linear equations in terms of variables t_(k). Each (ij) pair from the matrix is uniquely mapped to a one-dimensional vector. The collection of all these one-dimensional vectors form the coefficients matrix for the set of linear equations in terms of the variables t_(k).

A coefficient matrix is found that is square and nonsingular; hence, this set of linear equations is solved for variables of form “t_(k)”, block 145. In an alternative embodiment, a psuedoinverse (or best-fit linear solution) can be used to solve for variables t_(k). This alternative approach may include various rearrangement and/or elimination of rows and columns of the coefficient matrix for the t_(k) variables, as well as various decomposition methods for this matrix. Also, in at least one embodiment, the relinearization process may be reapplied to this latest set of quadratic equations to obtain a solution.

Referring now to block 150, the variables y_(k) are determined, and a solution set is defined to solve for power system unknowns represented by the set of non-linear equations, block 155. The dual transformation process disclosed above is referred to as relinearization. See “Crytpanalyis of the HFE Public Key Cryptosystem,” Kipnis and Shamir, published in the Proceedings of Crypto '99, Springer-Verlag, which presents a prior art example of 2^(nd)-order relinearization.

We now consider an example of solving a series of non-linear equations having 2^(nd)-order as well as linear terms, using the relinearization process, x ₁ ²−2x ₁ x ₂ +x ₃ ² +x ₂=8 2x ₂ x ₃ −x ₂ ²+2x ₁=10 the equations are presented as 5x ₁ x ₃ −x ₁ x ₂+3x ₃ +x ₂ ²=26 7x ₁ ²−2x ₂ x ₃ +x ₃ ²−5x ₃=−11 x ₁ ² −x ₂ ²−5x ₂ ₃+12x ₁=−13   (5)

Establishing a first transformation variable y_(ij) as y_(ij)=x_(i)x_(j) and assuming x₀−1, equations (5) are written as follows: $\begin{matrix} {{\begin{bmatrix} 1 & {- 2} & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & {- 1} & 2 & 0 & 0 \\ 0 & {- 1} & 0 & 0 & 0 & 1 & 0 & 5 & 3 \\ 7 & 0 & 1 & 0 & {- 2} & 0 & 0 & 0 & {- 5} \\ 1 & 0 & 0 & 0 & {- 5} & 1 & 12 & 0 & 0 \end{bmatrix}\begin{bmatrix} y_{11} \\ y_{12} \\ y_{33} \\ y_{20} \\ y_{23} \\ y_{22} \\ y_{10} \\ y_{13} \\ y_{30} \end{bmatrix}} = \begin{bmatrix} 8 \\ 10 \\ 26 \\ {- 11} \\ {- 13} \end{bmatrix}} & (6) \end{matrix}$

Equation 6 is then partitioned to form a non-singular square (left partition) and remaining matrix (right partition), with the variables associated with the right partition changed to z₁, z₂, etc.: $\begin{matrix} {{\begin{bmatrix} 1 & {- 2} & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & {- 1} & 2 & 0 & 0 \\ 0 & {- 1} & 0 & 0 & 0 & 1 & 0 & 5 & 3 \\ 7 & 0 & 1 & 0 & {- 2} & 0 & 0 & 0 & {- 5} \\ 1 & 0 & 0 & 0 & {- 5} & 1 & 12 & 0 & 0 \end{bmatrix}\begin{bmatrix} y_{11} \\ y_{12} \\ y_{33} \\ y_{20} \\ y_{23} \\ z_{1} \\ z_{2} \\ z_{3} \\ z_{4} \end{bmatrix}} = \begin{bmatrix} 8 \\ 10 \\ 26 \\ {- 11} \\ {- 13} \end{bmatrix}} & (7) \end{matrix}$

Further, variables “y” in Equation 7 are solved for in terms of the “z” variables. In large scale systems, various sparsity techniques and various matrix decomposition techniques may be used for computational efficiency.

In solving for y_(ij), each y_(ij) is now a linear function of the “z” variables. In the next step, all unique and existent possible combinations of two-tuple products (or pairs) of y_(ij) are formed, and a new set of quadratic equations in terms of the “z” variables is defined. These product pairs may be expressed in equations of the following form for all possible values of i, j, m, . . . , and n: y_(ij)y_(mn)=y_(im)y_(jn)= . . . =y_(in)y_(jm)   (8) The new set of quadratic equations shown in Equation 8 are solved through a second change of variable using, for example, z_(i)z_(i)=t_(k).

Continuing with the present example, a new nonsingular, 14×14 coefficient matrix is formulated as (equation (9)): $\begin{matrix} {{\begin{bmatrix} {- 241.5} & 2849 & 0 & 60 & 337 & 0 & 7.5 & 0 & {- 85} & 0 & {- 14.25} & {- 1989} & {- 1} & 0 \\ 64 & 0 & 260 & 156 & {- 17} & {- 10} & {- 6} & 0 & 0 & {- 30} & 0.5 & 0 & {- 25} & {- 9} \\ {- 163.5} & 1667 & {- 120} & {- 12} & 321 & {- 15} & {- 1.5} & 175 & 20 & 0 & {- 15} & {- 1700} & 0 & 0 \\ {- 13.5} & 97 & {- 26} & 0 & 10 & 1 & 0 & 0 & 0 & 3 & {- 0.75} & {- 17} & 5 & 0 \\ {- 90} & 10 & 0 & 0 & 118 & 0 & 5 & 0 & 0 & 0 & {- 9.75} & {- 1} & 0 & 0 \\ 0 & {- 85} & 0 & 0 & {- 9.5} & 0 & 0 & 0 & 5 & {- 1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 29 & 26 & 0 & 10 & {- 1} & {- 100} & 0 & {- 4} & 0 & 0 & 10 & 0 \\ 0 & 5 & 0 & 26 & 0.5 & 0 & {- 1} & 0 & 0 & {- 5} & 0 & {- 1} & 0 & {- 3} \\ {- 162} & 3042 & 430 & 385 & {- 117} & 48 & 23.5 & {- 586} & {- 351} & {- 25} & 9.5 & 0 & 0 & {- 15} \\ 1125.5 & {- 11893} & 850 & {- 55} & {- 2120} & 95 & {- 40} & {- 1170} & 382 & {- 50} & 95 & 11700 & 0 & {- 5} \\ 231 & {- 2600} & 115 & {- 61} & 101 & {- 60} & {- 31} & 500 & 300 & {- 35} & {- 10} & 0 & {- 50} & {- 3} \\ 8 & {- 26} & {- 25} & {- 15} & 1 & {- 1.5} & {- 1.5} & 5 & 3 & 0 & {- 0.5} & 0 & 0 & 0 \\ {- 64.5} & 529 & {- 50} & {- 5} & 60 & {- 5} & 0.5 & 10 & 1 & 0 & {- 5} & {- 100} & 0 & 0 \\ 0 & 0 & 0 & {- 12} & 0 & 0 & {- 1.5} & 1 & 17 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}\begin{bmatrix} t_{1} \\ t_{2} \\ t_{3} \\ t_{4} \\ t_{5} \\ t_{6} \\ t_{7} \\ t_{8} \\ t_{9} \\ t_{10} \\ t_{11} \\ t_{12} \\ t_{13} \\ t_{14} \end{bmatrix}} = \begin{bmatrix} 1020 \\ 676 \\ 348 \\ 60 \\ 25 \\ 0 \\ 0 \\ 0 \\ 2210 \\ {- 2465} \\ {- 754} \\ {- 130} \\ 145 \\ 0 \end{bmatrix}} & (9) \end{matrix}$

The t_(k) variables in equation 9 are then solved using known methods resulting in a “z” vector in the form of: [z₁z₂z₃z₄z₁z₂z₁z₃z₁z₄z₂z₃z₂z₄z₃z₄z₁ ²z₂ ²z₃ ²z₄ ²]=[4 1 3 3 4 12 12 3 3 9 16 1 9 9]  (10)

The y_(ij) values are then calculated using equation (3) as: Y=[y ₁₁ y ₁₂ y ₃₃ y ₂₀ y ₂₃ ]=[x ₁ ² x ₁ x ₂ x ₃ ² x ₂ x ₂ x ₃]=[1 2 9 2 6]  (11)

Substituting the values for “y” and “z”, and solving for “x”, the solution to the set of equations shown in equation 1 of this example is x₁=1,x₂=2, and x₃=3   (12)

Other varieties of this algorithm may be used by forming three-tuple (or even higher) product combinations such as: y_(ij)y_(mn)y_(pq)=y_(pi)y_(qm)y_(jn)=y_(pn)y_(jm)y_(qi)= . . . =y_(qn)y_(pm)y_(i)   (13)

Considering now yet another example of the non-iterative approach, the method described herein is extended to solve a set of third-order nonlinear equations. Consider the following set of equations: x ₁ ³+3x ₁ ² x ₂ +x ₂ ²=11 5x ₁ x ₂ ² +x ₂ ³−3x ₁=25 x ₁ x ₂ x ₃ +x ₂ x ₃ +x ₁ −x ₃ ³=−14 5x ₁ ² x ₂ −x ₂ x ₃ −x ₂ ²=0 x ₂ ³ +x ₁ ² x ₂ +x ₁ x ₂ x ₃+2x ₁=18 x ₁ ³ +x ₂ ² +x ₂ x ₃ +x ₁=12 −2x ₁ ³+5x ₂ ³ −x ₁ x ₂ ³ +x ₁ x ₂ x ₃ −x ₃ ³=21   (14)

Once again, establishing a first transformation variable y_(ijk) as y_(ijk)=x_(i)x_(j)x_(k) and assuming x₀=1, the above equations are written in a non-singular square (left partition) and remaining matrix (right partition) form as follows: $\begin{matrix} {{\begin{bmatrix} 1 & 0 & 0 & 3 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 3 & 0 & 0 & 5 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & {- 1} \\ 0 & 0 & 0 & 5 & {- 1} & 0 & 0 & {- 1} & 0 \\ 0 & 1 & 2 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ {- 2} & 5 & 0 & 0 & 0 & 1 & 1 & 0 & {- 1} \end{bmatrix}\begin{bmatrix} y_{111} \\ y_{222} \\ y_{100} \\ y_{112} \\ y_{220} \\ y_{122} \\ y_{123} \\ z_{1} \\ z_{2} \end{bmatrix}} = \begin{bmatrix} 11 \\ 25 \\ {- 14} \\ 0 \\ 18 \\ 12 \\ 21 \end{bmatrix}} & (15) \end{matrix}$

Variables “y” in Equation 15 are determined in terms of the “z” variables, where each y_(ijk) is now a linear function of the “z” variables. In a next step, unique and existent possible combinations of two-tuple products (or pairs) of y_(ijk) are formed resulting in a set of quadratic equations in terms of the z variables. These equations, resulting from equal product pairs, may be expressed in the following form for all possible values of i, j, k,m, n, . . . , and l: y_(ijk)y_(mnl)=y_(imk)y_(jnl)= . . . =y_(inl)y_(jmk)   (16)

The new set of quadratic equations shown in Equation 16 are solved through a second change of variable, using e.g. z_(i)z_(j)=t_(k). $\begin{matrix} {{\begin{bmatrix} 1222.6 & {- 927.6} & {- 69.9} & 462 & 26.5 \\ 1245.9 & {- 972.8} & {- 61.8} & 39.3 & 24.2 \\ {- 539} & 432.7 & 26.6 & {- 16.8} & {- 10.6} \\ 874.2 & {- 719.8} & {- 43.3} & 26.2 & 17.9 \\ 71.5 & {- 4.3} & {- 3.3} & 3.8 & 0 \end{bmatrix}\begin{bmatrix} t_{1} \\ t_{2} \\ t_{3} \\ t_{4} \\ t_{5} \end{bmatrix}} = \begin{bmatrix} {- 8068.9} \\ {- 9739.0} \\ 4417.7 \\ {- 7253.3} \\ {- 80.7} \end{bmatrix}} & (17) \end{matrix}$

The t_(k) variables in equation 17 are then solved using known methods resulting in a “z” vector in the form of: [z₁z₂z₁z_(2=l z) ₁ ²z₂ ²]=6 27 162 36 729]  (18)

Of note, when a direct solution of the linear set of equations is not possible, a pseudo-inverse or best-fit linear solution may be used to solve for t_(k). This may include various rearrangement and/or elimination of rows and columns of the coefficient matrix for the t_(k) variables, as well as various decomposition methods for this matrix.

The y_(ijk) values are then calculated using equation (3) and shown to be $\begin{matrix} {Y = {\begin{bmatrix} y_{111} & y_{222} & y_{100} & y_{112} & y_{220} & y_{112} & Y_{123} \end{bmatrix}\quad = {\begin{bmatrix} x_{1}^{3} & x_{2}^{3} & x_{1} & {x_{1}^{2}x_{2}} & x_{2}^{2} & {x_{1}x_{2}^{2}} & {x_{1}x_{2}x_{3}} \end{bmatrix}\quad = \begin{bmatrix} 1 & 8 & 1 & 2 & 4 & 4 & 6 \end{bmatrix}}}} & (19) \end{matrix}$

Hence, the solution to the set of equations shown in equation 19 is shown to be: x₁−1, x₂=2, and x₃=3   (20)

Again, as in the quadratic case, other varieties of this algorithm may be used by forming three-tuple (or even higher) product combinations such as: y_(ijk)y_(mnl)y_(pqr)=y_(pik)y_(qml)y_(jnr)=y_(pnk)y_(jml)y_(qir)= . . . =y_(qnk)y_(pml)y_(ijr)   (21) It should be noted that, without loss of generality, these examples are created to have integer valued solutions. In general the solution can be any real value. Iterative Solution Method

FIG. 2 illustrates a flow chart 200 of an exemplary process in accordance with an embodiment. In this embodiment, system parameters and topology data are read, made available or obtained, block 205. Since there may be no direct synchronized measurements available, initial values for the unknowns such as voltage phasors, magnitudes, and/or phase angles may be estimated or assumed, block 210.

Considering further iterative method of FIG. 2, a plurality of non-linear power flow equations are reconfigured to form a set of quadratic equations, a variables vector and a right hand side (RHS) vector, block 220. The non-linear equations are placed in a matrix format to take advantage of rapid processing by electronic means, such as a computer or parallel operations by a plurality of computers.

Still referring to FIG. 2, as shown in block 225, the matrix representation of the non-linear equations is parsed as previously discussed with the non-iterative approach. To conform to the parsed matrix, the variables vector is divided into a plurality of “y” and “z” variables (block 230), similar to the operation performed at block 120 in FIG. 1. In block 235, the “y” variable terms are computed in terms of the “z” variables, similar to the operation performed in block 125 of FIG. 1. Further, in block 240, a process to obtain unique variable pair combinations is executed, and, in block 245, a set of linear equations is formulated. A change of variable (block 250) is then performed. Once the linear equations have been defined, the equations are solved for “z” (block 255), and consequently values of “y” are determined, block 260.

In the following step, a determination is made whether the computed values are converging to a solution, block 265. If the answer is negative, then the estimated voltage values are updated in block 270, and the iterative process continues, block 220.

However, if the values are converging to a solution, then the converging voltages are made available for recording or display, for example, in block 275. The criteria for convergence may be satisfied when a difference between a solution set from a current iteration and a solution set from a previous iteration is within a known tolerance limit. In at least one embodiment, the criteria for convergence may be satisfied when the difference between the present solution set and the previous solution set is within 0.1 percent.

The iterative method shown in FIG. 2 is applicable when a sufficient number of actual measurements is not available, including the case where no measurements are available. When a sufficient number of actual measurements are not available, an initial estimation of the variable values is used. In an AC power system, represented by “E ” non-linear equations requiring the determination of “E” unknowns, a solution may be determined using a non-iterative process when a sufficient number of measurements exists, i.e., when the number of measurements is in the order of (depending on the topology or interconnectivity of the transmission network being analyzed): $\begin{matrix} {{{\frac{E}{2}\quad{when}\quad E\quad{is}\quad{even}};}{\frac{E - 1}{2}\quad{when}\quad E\quad{is}\quad{odd}}} & (22) \end{matrix}$

To apply the steps of FIG. 2 to AC power flow analysis, three different problem formulations may be used as described below, specifically: real-variable equations in polar form; real-variable equations in rectangular form; and complex-variable equations. Also, a formulation for the AC state estimation problem is also provided showing the application of the above steps.

Formulation 1: Real-Variable Equations in Polar Form

Complex power flow equations for an N-bus power system are divided into real and reactive power equations to form the following set of 2( N-1) real equations: $\begin{matrix} {{{{\sum\limits_{i = l}^{N}{V_{i}V_{j}V_{ij}{\cos\left( {\delta_{i} - \delta_{j} + \theta_{ij}} \right)}}} = {P_{G_{j}} - P_{D_{j}}}};{{{and} - {\sum\limits_{i = l}^{N}{V_{i}V_{j}V_{ij}{\sin\left( {\delta_{i} - \delta_{j} + \theta_{ij}} \right)}}}} = {Q_{G_{j}} - Q_{d_{j}}}}}{{j = 2},{\ldots\quad N}}} & (23) \end{matrix}$ where:

-   -   V_(i) represents the voltage magnitude at bus i.     -   δ_(i) represents the voltage phase-angle at bus i.     -   Y_(ij), Y_(ii) represent magnitudes of complex bus admittance         matrix elements     -   θ_(ij) represents the angle of Y_(ij), the complex admittance         quantities in the system bus admittance (Ybus) matrix     -   P_(Dj), Q_(Dj) represent real and reactive demand at bus j     -   P_(Gj), Q_(Gj) represent real and reactive voltage generation at         bus j

The first bus, i.e., bus number one, is designated as the slack or swing bus, and voltage at this bus is assumed to be known. In one embodiment, the slack bus voltage may be represented as a normalized value of 1 per unit at an angle of zero degrees. Therefore, the equations for the state estimation are written for buses 2 to N. The slack bus provides the system losses, as well as any additional power required to maintain total system power balance.

In at least one embodiment, when direct synchronous measurement of the bus voltage phase-angles is available, the sine and cosine terms associated with the phase angles may be directly evaluated using the measured angles. The equations may then be expressed in terms of the products of bus voltage magnitudes. When some, but not all, of the bus voltage magnitudes are also known, then some linear terms will also exist in the problem formulation. All sine and cosine terms above may be replaced by numerical values, and the equations will be in quadratic form containing second order V_(i) V_(j) and/or linear terms V_(i), conforming to the form required to obtain a solution using the method described herein. The routines and algorithms developed herein will transform Equation 23 into the quadratic form and, hence, into a form solvable using the relinearization technique described above.

However, when a sufficient number of measurements of bus voltage phase-angles is not available, a DC power flow solution, known in the art, may be obtained to acquire a first approximate set of values for the bus voltage phase-angles.

The estimated values obtained from the DC power flow solution may then be used to determine sine and cosine terms, thus rendering equation 23 in a desired form. In this instance, the first iteration results provide an approximate one-shot solution to the AC power flow problem. The initial bus voltage phase-angles obtained can be iteratively updated, and equations re-solved via relinearization until a solution satisfying a desired convergence criterion is achieved.

Formulation 2: Real-Variable Equations in Rectangular Form

In this formulation, the complex bus voltage phasors are represented in rectangular form, i.e., {overscore (V)}i=a_(i)+jb_(i), and the power flow equations for an N-bus power system may be formulated as 2(N-1) real equations as: $\begin{matrix} {{{{{\sum\limits_{i = l}^{N}{{{Re}\left( Y_{ij} \right)} \cdot \left( {{a_{j}a_{i}} + {b_{j}b_{i}}} \right)}} + {{{Im}\left( Y_{ij} \right)} \cdot \left( {{a_{i}a_{j}} - {a_{j}b_{i}}} \right)}} = {P_{G_{j}} - P_{D_{j}}}};{{{and}\text{} - {\sum\limits_{i = l}^{N}{{{Re}\left( Y_{ij} \right)} \cdot \left( {{a_{j}a_{i}} - {a_{i}b_{j}}} \right)}} + {{{Im}\left( Y_{ij} \right)} \cdot \left( {{a_{i}a_{j}} = {b_{j}b_{i}}} \right)}} = {Q_{G_{j}} - Q_{D_{j}}}}}{{j = 2},{\ldots\quad N}}} & (24) \end{matrix}$

Using this approach, the formulation produces product terms of the real and imaginary parts of bus voltage phasor variables. If a voltage phasor is known, then linear terms will also appear in the equations. This formulation is therefore suitable for application of the method described herein.

Formulation 3: Complex-Variable Equations

In this formulation, a set of (N-1) equations in (N-1) voltage phasor variables is formed with all variables retained in the complex form. In this instance, the bus admittance matrix elements {overscore (Y)}ij are known and the equations (25) are in the desired form. However, the variables are complex quantities. The formulation is as follows: $\begin{matrix} {{{\sum\limits_{i = l}^{N}{{\overset{\_}{V}}_{i}{\overset{\_}{V}}_{j}^{*}{\overset{\_}{Y}}_{ij}}} = {S_{G_{j}} - S_{D_{j}}}}{{j = 2},{\ldots\quad N}}} & (25) \end{matrix}$ where:

-   -   {overscore (V)}_(i) represents the voltage phasor value at bus         i.     -   {overscore (Y)}_(ij) represents the complex bus admittance         matrix element values;     -   S_(D)=P_(D)+jQ_(D) represents the complex demand or load at a         given bus;     -   P_(D) and Q_(D) represent the real and reactive demand at the         bus, respectively;     -   S_(G)=P_(G)+jQ_(G) represents the complex power generation at a         given bus; P_(G) and Q_(G) represent the real and reactive         demand at the bus, respectively; and         -   the superscript * indicates the conjugate of the complex             value.

As an example of an operational application of the method described herein for solving the AC power flow problem, consider the following 4-bus power system shown in FIG. 3. Illustratively, three generators or sources 310, 312, and 314, supply power through buses 320, 322, 324, and 326, respectively, in network 300. The source (generation) data and transmission line and bus load data for this system are shown in Tables 1 and 2, respectively. TABLE 1 Bus Data Bus Voltage Bus Load Bus Generation Bus. Bus Magnitude Phase-angle Mega Mega No. Code Per Unit Deg. watts Megavars watts Megavars 1 1 1.0 0.0 0.0 0.0 0.0 0.0 2 0 — — 0.0 0.0 0.0 0.0 3 0 — — 2800 400 1000 200 4 0 0.927 −18.038 2000 250 500 50

With regard to Table 1, column 1 represents a bus number and column 2 represents a bus code indicating the type of bus. In particular, as is known in the art, three types of buses and the associated codes can exist in power networks. A bus code equal to one (1) indicates a slack or swing bus. Further, a bus code equal to zero (0) indicates a load bus where injected MW(Megawatts) and Mvar are known, and the voltage phasor values may be determined using the power flow solution described herein. A third kind of bus, which is not used in this example, represents a voltage-control bus, wherein the magnitude of the bus voltage is known and the MVAR injection at the bus is determined via the power flow solution. In this case, the MVAR injection becomes an unknown creating a linear term in the set of equations, for which a solution exists. Still referring to Table 1, columns 3 and 4 show an initial value for the magnitude and phase of the voltage at bus 1, i.e., the slack bus, and for any other bus for which the values may be directly measured. The voltages at busses 2-3 are unknown and are determined using the method described herein. Columns 5 and 6 of Table 1 indicate the load and generation levels, respectively, at each bus on a per unit (p.u.) basis. TABLE 2 Transmission Line Data Sending Receiving ½ Shunt End End Susceptance (Bc) Bus No. Bus No. R(p.u.) X(p.u.) (p.u.) Line 1 2 0.00100 0.00600 1.0 No. 1 2 3 0.00200 0.02023 1.0 2 2 4 0.00300 0.02110 1.0 3 1 3 0.00500 0.04000 1.0 4 1 4 0.00500 0.04000 1.0 5 3 4 0.00500 0.04000 1.0

Table 2 provides the transmission line data for the simple power system presented in FIG. 3. In Table 2, column 1 represents the line number; columns 2 and 3 represent the bus numbers of the two ends of each line; columns 3 to 5 provide the “pi” model parameters of each transmission line. These parameters consist of the series resistance and reactance as well as the shunt susceptance values per unit.

In particular, the AC power flow equations may be formulated in real-variable rectangular form as follows: $\begin{matrix} {{{\sum\limits_{i = l}^{4}{{{Re}\left( Y_{i2} \right)} \cdot \left( {{a_{2}a_{I}} + {b_{2}b_{i}}} \right)}} + {{{Im}\left( Y_{i2} \right)} \cdot \left( {{a_{i}b_{2}} - {a_{2}b_{i}}} \right)}} = 0} & (26) \\ {{{- {\sum\limits_{i = l}^{4}{{{Re}\left( Y_{i2} \right)} \cdot \left( {{a_{2}b_{i}} - {a_{i}b_{2}}} \right)}}} + {{{Im}\left( Y_{i2} \right)} \cdot \left( {{a_{i}a_{2}} + {b_{2}b_{i}}} \right)}} = 0} & \quad \\ {{{\sum\limits_{i = l}^{4}{{{Re}\left( Y_{i3} \right)} \cdot \left( {{a_{3}a_{i}} - {b_{3}b_{i}}} \right)}} + {{{Im}\left( Y_{i3} \right)} \cdot \left( {{a_{i}b_{3}} + {a_{3}b_{i}}} \right)}} = {10 - 28}} & (27) \\ {{{- {\sum\limits_{i = l}^{4}{{{Re}\left( Y_{i3} \right)} \cdot \left( {{a_{3}b_{i}} - {a_{i}b_{3}}} \right)}}} + {{{Im}\left( Y_{i3} \right)} \cdot \left( {{a_{i}a_{3}} + {b_{3}b_{i}}} \right)}} = {2 - 4}} & \quad \\ {{{\sum\limits_{i = l}^{4}{{{Re}\left( Y_{i4} \right)} \cdot \left( {{a_{4}a_{i}} + {b_{4}b_{i}}} \right)}} + {{{Im}\left( Y_{i4} \right)} \cdot \left( {{a_{i}b_{4}} - {a_{4}b_{i}}} \right)}} = {5 - 20}} & (28) \\ {{{- {\sum\limits_{i = l}^{4}{{{Re}\left( Y_{i4} \right)} \cdot \left( {{a_{4}b_{i}} - {a_{i}b_{3}}} \right)}}} + {{{Im}\left( Y_{i4} \right)} \cdot \left( {{a_{i}a_{4}} + {b_{4}b_{i}}} \right)}} = {0.5 - 2.5}} & \quad \end{matrix}$ where:

-   -   {overscore (V)}_(i)=a_(i)+jb_(i); and     -   Re (Y_(ij)) and Im (Y_(ij)) are the real and imaginary parts, of         the complex     -   Y_(bus) matrix elements for this system formed by utilizing the         transmission line parameters given in Table 2; and the right         hand side values are the net injection (generation minus load)         real and reactive power values in per-unit, obtained by dividing         the actual values in Table 1 by the base power value which is         100 MVA (mega-volt-amperes).

Furthermore, in this case, the voltage at bus 4 is treated as measured or known (as shown in Table 1). Therefore a sufficient number of measurements or known voltages (buses 1 and 4) are available, and the solution is determined using the non-iterative approach.

Table 3 represents the results of a power flow solution showing the determined voltage phasor values at busses 2 and 3 under the given loading condition, in accordance with the principles of the present disclosure. TABLE 3 Bus Voltage Bus Load Bus Generation Bus. Bus Magnitude Phase-angle Mega Mega No. Code Per Unit Deg. watts Megavars watts Megavars 1 1 1.0 0.0 0.0 0.0 3448.904 445.52 2 0 0.968 −6.602 0.0 0.0 0.0 0.0 3 0 0.924 −19.464 2800 400 1000 200 4 0 0.927 −18.038 2000 250 500 50

Alternatively, if the information regarding the voltage on a given bus, for example bus 4, were not available, the solution to the voltages on busses 2, 3 and 4 may be obtained by iteratively applying the principles disclosed herein until a desired convergence criterion is satisfied. After the voltage on each bus is determined, the power flow levels in all transmission lines, system losses may be easily computed using techniques well known in the art. The disclosed methodology has also been applied to the IEEE 30-bus benchmark power flow case and the correct solution is obtained.

State-Estimation (SE) Formulation

With an SE tool, generally, three types of SCADA measurements are used, and an appropriate set of equations are written for each type of measurement. Each measurement may have an error, and can be treated as an independent, zero-mean, Gaussian distributed random variable described with its probability density function. These measurement equations include: 1) Bus power equations utilized when the (injected) bus power quantities are measured; 2) Bus voltage magnitude equations; and 3) Transmission line real and reactive flow equations when they are measured. An objective function in the least-squares sense is formed to determine the best set of system states that fit the measured data.

The SE problem, using rectangular variables {overscore (V)}i=ai+jbi, is formulated as a maximum likelihood weighted Least Squares problem as follows: $\begin{matrix} {{{Min}.\quad{J\left( \underset{\_}{x} \right)}} = {\sum\limits_{i = 1}^{M}\frac{\left\lbrack {Z_{i} - {f_{i}\left( \underset{\_}{x} \right)}} \right\rbrack^{2}}{\sigma_{i}^{2}}}} & (29) \end{matrix}$ where:

-   -   M is the number of measurements;     -   J is the error function or the residual error;     -   z_(i) is the i^(th) measurement value;     -   x is the a vector containing all a_(i) s and b_(i) s;     -   f_(i) is the function used to calculate the value being measured         by the i^(th) measurement; and     -   σ_(i) ² is a constant representing the variance for the i^(th)         measurement

Each term in the summation above has a form depending on the quantity being measured. For example, for voltage magnitude, the corresponding term in equation 26 may be represented as: $\begin{matrix} \frac{\left\lbrack {V_{i}^{{meas}^{2}} - \left( {a_{i}^{2} + b_{i}^{2}} \right)} \right\rbrack^{2}}{\sigma_{v_{i}}^{2}} & (30) \end{matrix}$

Real and reactive power flows on line ij, connecting buses i and j, may be represented as: $\begin{matrix} {\frac{\begin{matrix} \left\lbrack {{MW}_{ij}^{meas} - {\left( {a_{i}^{2} + b_{i}^{2}} \right){\Re\left( y_{ij} \right)}} +} \right. \\ \left. {{\left( {{a_{i}a_{j}} + {b_{i}b_{j}}} \right){\Re\left( y_{ij} \right)}} - {\left( {{a_{i}b_{j}} - {a_{j}b_{i}}} \right){{\mathfrak{J}}\left( y_{ij} \right)}}} \right\rbrack^{2} \end{matrix}}{\sigma_{{MW}_{ij}^{2}}};{and}} & (31) \\ \frac{\begin{matrix} \left\lbrack {{MVAR}_{ij}^{meas} + {\left( {a_{i}^{2} + b_{i}^{2}} \right){{\mathfrak{J}}\left( {y_{ij} + {Bc}} \right)}} -} \right. \\ \left. {{\left( {{a_{i}a_{j}} + {b_{i}b_{j}}} \right){{\mathfrak{J}}\left( y_{ij} \right)}} + {\left( {{a_{i}b_{j}} - {a_{j}b_{i}}} \right){\Re\left( y_{ij} \right)}}} \right\rbrack^{2} \end{matrix}}{\sigma_{{MVAR}_{ij}^{2}}} & (32) \end{matrix}$ where:

-   -   y_(ij) is the complex value of the series admittance of the         transmission line connected between buses i and j; and     -   Bc is half of the shunt susceptance of that line; and     -   and ℑ signify the real and imaginary parts of a complex number,         respectively.

For bus power measurements, the equations are identical to the power flow equations (24) above. Similar equations may be derived and written for line current measurements when line current measurements are available.

To minimize the above objective function J(x), its derivatives with respect to each variable are forced to zero. Each derivative equation in this case will be a polynomial of degree three or less. For example, the derivatives for typical voltage magnitude terms in J are in the form: $\begin{matrix} {\frac{\partial J_{v}}{\partial a_{i}} = {\frac{{- 4}{a_{i}\left\lbrack {V_{i}^{{meas}^{2}} - \left( {a_{i}^{2} + b_{i}^{2}} \right)} \right\rbrack}}{\sigma_{v_{i}}^{2}} = 0}} & (33) \end{matrix}$ and the derivatives for typical real and reactive power flow terms in J are in the form: $\begin{matrix} {\frac{\begin{matrix} {2\left\lbrack {{MW}_{ij}^{meas} - {\left( {a_{i}^{2} + b_{i}^{2}} \right){\Re\left( y_{ij} \right)}} + {\left( {{a_{i}a_{j}} + {b_{i}b_{j}}} \right){\Re\left( y_{ij} \right)}} -} \right.} \\ {\left. {\left( {{a_{i}b_{j}} - {a_{j}b_{i}}} \right){{\mathfrak{J}}\left( y_{ij} \right)}} \right\rbrack\left\lbrack {{\left( {{{- 2}a_{i}} + a_{j}} \right){\Re\left( y_{ij} \right)}} - {b_{j}{{\mathfrak{J}}\left( y_{ij} \right)}}} \right\rbrack} \end{matrix}}{\sigma_{{MWAR}_{ij}^{2}}} = 0} & (34) \\ {\frac{\partial J_{q}}{\partial a_{i}} = 0} & \quad \\ {\frac{\begin{matrix} {2\left\lbrack {{MVAR}_{ij}^{meas} - {\left( {a_{i}^{2} + b_{i}^{2}} \right){{\mathfrak{J}}\left( {y_{ij} + {Bc}} \right)}} -} \right.} \\ \left. {{\left( {{a_{i}a_{j}} + {b_{i}b_{j}}} \right){{\mathfrak{J}}\left( y_{ij} \right)}} + {\left( {{a_{i}b_{j}} - {a_{j}b_{i}}} \right){\Re\left( y_{ij} \right)}}} \right\rbrack \\ \left\lbrack {{2a_{i}{{\mathfrak{J}}\left( {y_{ij} + {Bc}} \right)}} - {a_{j}{{\mathfrak{J}}\left( y_{ij} \right)}} + {b_{j}{\Re\left( y_{ij} \right)}}} \right\rbrack \end{matrix}}{\sigma_{{MWAR}_{uj}^{2}}} = 0} & (35) \end{matrix}$

Derivatives with respect to the b_(i) variables may be similarly written. These third-order equations are in a form suitable for solution using the third order extension of the solution method described herein.

Optimal Power Flow (OPF) formulation

In OPF problems, an objective function is formed and minimized (or maximized) subject to power flow equations as equality constraints. Inequality constraints can also be accommodated via various methods known in the art. The goal is to find a solution or operating point for the power system such that the objective is satisfied. For example, the objective can be minimization of transmission system losses, maintenance of a certain voltage profile among a number of buses, or minimization of control effort (such as MVAR injection into a number of buses) expenditure. A combination of objectives can also be bundled together to form a single objective function as well.

An example of the OPF problem formulation for an N-bus power system based on the real-variable power flow equations is presented below: $\begin{matrix} {{{Min}.\quad{g\left( \underset{\_}{x} \right)}} = {\sum\limits_{i = 1}^{w}\left\lbrack {V_{i}^{{des}^{2}} - \left( {a_{i}^{2} + b_{i}^{2}} \right)} \right\rbrack^{2}}} & (36) \end{matrix}$ Subject to: $\begin{matrix} {{{{{\sum\limits_{i = 1}^{N}{{{Re}\left( Y_{ij} \right)} \cdot \left( {{a_{j}a_{i}} + {b_{j}b_{i}}} \right)}} + {{{Im}\left( Y_{ij} \right)} \cdot \left( {{a_{i}b_{j}} - {a_{j}b_{i}}} \right)}} = {P_{G_{j}} - P_{D_{j}}}};{{{and} - {\sum\limits_{i = 1}^{N}{{{Re}\left( Y_{{ij}\quad} \right)} \cdot \left( {{a_{j}b_{i}} + {a_{i}b_{j}}} \right)}} + {{{Im}\left( Y_{ij} \right)} \cdot \left( {{a_{i}a_{j}} - {b_{j}b_{i}}} \right)}} = {q_{j} + Q_{G_{j}} - Q_{D_{j}}}}}{{j = 2},{\ldots\quad N}}} & (37) \end{matrix}$ where:

-   -   w is the number of buses where a desired voltage is to be         maintained;     -   g is the objective function to be minimized     -   x is the vector containing all a_(i) s and b_(i) s and the         additional     -   MVAR injection control variables q_(i); and     -   v_(i) ^(des) is the desired voltage magnitude to be maintained         at the i^(th) bus.

Generally, for solving this type of optimization problem, a Lagrange function is formed by introducing Lagrange multipliers λ_(k) to incorporate the constraint equations in the objective function as follows: $\begin{matrix} {{L\left( {\underset{\_}{x},\underset{\_}{\lambda}} \right)} = {{g\left( \underset{\_}{x} \right)} - {\lambda_{1}\left\lbrack {{\sum\limits_{i = 1}^{N}\quad{{{Re}\left( Y_{i2} \right)} \cdot \left( {{a_{2}a_{i}} + {b_{2}b_{i}}} \right)}} + {{{Im}\left( Y_{i2} \right)} \cdot \left( {{a_{i}b_{2}} - {a_{2}b_{i}}} \right)} - P_{G_{2}} + P_{D_{2}}} \right\rbrack} - {\ldots\quad{\lambda_{N - 1}\left\lbrack {{\sum\limits_{i = 1}^{N}\quad{{{Re}\left( Y_{iN} \right)} \cdot \left( {{a_{N}a_{i}} + {b_{N}b_{i}}} \right)}} + {{{Im}\left( Y_{iN} \right)} \cdot \left( {{a_{i}b_{N}} - {a_{N}b_{i}}} \right)} - P_{G_{N}} + P_{D_{N}}} \right\rbrack}} - {\ldots\quad{\lambda_{N}\left\lbrack {{\sum\limits_{i = 1}^{N}\quad{{{Re}\left( Y_{i2} \right)} \cdot \left( {{a_{2}b_{i}} + {b_{2}a_{i}}} \right)}} + {{{Im}\left( Y_{i2} \right)} \cdot \left( {{a_{i}a_{2}} - {b_{2}b_{i}}} \right)} - q_{2} - Q_{G_{2}} + Q_{D_{2}}} \right\rbrack}} - {\ldots\quad{\lambda_{2{({N - 1})}}\left\lbrack {{\sum\limits_{i = 1}^{N}\quad{{{Re}\left( Y_{iN} \right)} \cdot \left( {{a_{N}b_{i}} + {b_{N}a_{i}}} \right)}} + {{{Im}\left( Y_{iN} \right)} \cdot \left( {{a_{i}a_{N}} - {b_{n}b_{i}}} \right)} - q_{N} - Q_{G_{N}} + Q_{D_{N}}} \right\rbrack}}}} & (38) \end{matrix}$

Partial derivatives of L with respect to elements of x and λ vectors are taken and forced to zero to obtain the optimal solution. As can be seen from the above equations, L is a polynomial of degree 4, and the derivatives will be of degree three or less. Once again these equations can be solved using the methodologies developed and disclosed in this application.

Other objective functions such as system loss minimization or a combination of various objectives can be formed. Inequality constraints can also be similarly handled via known methods such as barrier functions, etc.

Economic Dispatch and Unit Commitment

Economic dispatch is the problem of determining the most economic or least-cost output levels of a number of generators required to supply a total given load. The cost curves for each unit are traditionally described via quadratic functions such as: C _(i)(P _(G) _(i) )=α_(i) P _(G) _(i) ²+β_(i) P _(G) _(i) +δ_(i)   (39) where C_(i) is the cost in $/MW of running unit i at P_(Gi) MW output.

To determine the minimum cost operation of the system, a constrained minimization problem is formulated as follows: $\begin{matrix} \begin{matrix} {{{Min}.{\sum\limits_{i = 1}^{U}\quad{\alpha_{i}P_{G_{i}}^{2}}}} + {\beta_{i}P_{G_{i}}} + \delta_{i}} \\ {{{Subject}\quad{{to}:{\sum\limits_{i = 1}^{U}\quad P_{G_{i}}}}} = {P_{D} + P_{Loss}}} \end{matrix} & (40) \end{matrix}$ where:

-   -   U is the total number of generators to be dispatched;     -   P_(D) is a constant indicating total MW demand in the system;         and     -   P_(LOSS) is the total transmission losses expressed as a         quadratic function of the     -   P_(G)S using the well known B-coefficients as: $\begin{matrix}         {P_{Loss} = {\sum\limits_{m = 1}^{U}\quad{\sum\limits_{n = 1}^{U}\quad{P_{G_{m}}B_{mn}P_{G_{n}}}}}} & (41)         \end{matrix}$         which may also include linear and constant terms.

To solve this constrained optimization problem, again a Lagrange Function is formed using a Lagrange multiplier as: $\begin{matrix} {{L\left( {\underset{\_}{P_{G}},\lambda} \right)} = {\left( {{\sum\limits_{i = 1}^{U}\quad{\alpha_{i}P_{G_{i}}^{2}}} + {\beta_{i}P_{G_{i}}} + \delta_{i}} \right) - {\lambda\left\lbrack {{\sum\limits_{i = 1}^{U}\quad P_{G_{i}}} - P_{D} - P_{Loss}} \right\rbrack}}} & (42) \end{matrix}$

Partial derivatives of the Lagrange function L with respect to all P_(Gi) and λ are forced to zero to obtain the optimum or minimum-cost solution. As can be seen from the above formulation, the partial derivative equations will all be of second degree or less, and again the problem can be solved using the methodologies described herein.

Note that the Unit Commitment or Optimal Scheduling problem is similar to the ED problem. However, the scheduling is performed not for a given fixed value of P_(D), but for a load profile curve indicating the system load over a period of time ranging from a few hours to a day or more. In solving this problem, the load profile curve is approximated with a set of fixed demand intervals, for example one hour each. The optimization problem is solved over the entire period of the load profile considered. In each interval, the ED dispatch problem may be solved to determine the least-cost output levels of the generators for many “on-off” generator combinations. The above ED solution algorithm is therefore directly applicable.

Another type of Unit Commitment problem which is solved at the utility and/or Independent System Operator (ISO) control centers is the Security Constrained Unit Commitment (SCUC). This problem is similar in nature to the Unit Commitment problem discussed above. However, in each interval DC or AC power flow and/or stability solutions are performed to ensure steady-state and angle and voltage stability of the generating schedule being determined, considering a set of contingencies. With existing power flow solution methods, running an AC power flow solution in each interval, for all the generating unit combinations, is prohibitive and usually not attempted. The methodologies developed herein provide SCUC problem solving considering voltage stability concerns directly, by repeatedly and speedily computing the required AC power flow solutions.

Considering now FIG. 4, a system 400 for implementing the principles disclosed herein—such as the processes of FIGS. 1 or 2—is shown. In this exemplary system 400, input data is received from sources 402 over network 404, and is processed in accordance with one or more software programs executed by processing system 406. The results of processing system 406 may then be displayed on display 408, reported on reporting device 410, and/or processed by a second processing system 412. In system 400, the results of the processing system 406 are transmitted over network 414 to one or more of the display 408, the reporting system 410 or the processing system 412.

Specifically, processing system 406 includes one or more input/output devices 416 that receive data from the illustrated source devices 402 over network 404. The received data is then applied to processor 418, which is in communication with input/output device 416 and memory 420. Input/output device 416, processor 418 and memory 420 may communicate over a communication medium 422. Communication medium 422 may represent a communication network, e.g., ISA, PCI, PCMCIA bus, one or more internal connections of a circuit, a circuit card or other device, as well as portions and combinations of these and other communication media. Processing system 406 or processor 418 may be representative of a handheld calculator, special purpose or general purpose processing system, desktop computer, laptop computer, palm computer, or personal digital assistant (PDA) device, etc., as well as portions or combinations of these and other devices that can perform the operations illustrated in FIGS. 1 and 2.

In one embodiment, processor 418 includes software which, when executed, performs the operations illustrated herein. The software is contained in memory 420, is read or downloaded from a memory medium such as a CD-ROM or floppy disk 424, is provided by manual input device 426, such as a keyboard or a keypad entry, or is read from a magnetic or optical medium 428 when needed. Information items provided by memory medium 424, or input device 426, or a magnetic medium 428 are accessible to processor 418 through input/output device 416. Further, the data received by input/output device 416 may be immediately accessible by processor 418, or may be stored in memory 420. Processor 418 may further provide the results of the processing shown herein to display 408, reporting device 410, or a second processing unit 412 through I/O device 416.

As one skilled in the art recognizes, the terms processor, processing system, computer or computer system may represent one or more processing units in communication with one or more memory units and other devices, e.g., peripherals, connected electronically to and communicating with the at least one processing unit. Furthermore, the devices illustrated may be electronically connected to the one or more processing units via internal busses, e.g., ISA bus, micro channel bus, PCI bus, PCMCIA bus, etc. Alternatively, they may be connected via one or more internal connections of a circuit, circuit card or other device, as well as portions and combinations of these and other communication media, or an external network, e.g., the Internet and Intranet. In other embodiments, hardware circuitry (e.g., a VLSI integrated circuit) may be used in place of, or in combination with, software instructions to implement the embodiments disclosed herein. For example, the elements illustrated herein may also be implemented as discrete hardware elements or may be integrated into a single unit.

FIG. 5 shows a process 499 for non-iterative or iterative non-gradient power system analysis, in accord with one embodiment. Process 499 may be used to analyze a power system and may be implemented by a computer system or processor, such as processor 418 in FIG. 4. As shown in FIG. 5, a set of non-linear equations representative of power system unknowns is formulated, block 500. The set of non-linear equations are transformed into a corresponding first set of quadratic equations, block 502. In block 504, a first change of variable operation is performed, and the first set of quadratic equations is transformed into a first set of linear equations. The first set of linear equations is solved, parametrically, (block 506), thereby defining a second set of quadratic equations. Performing a second variable change, the second set of quadratic equations is transformed into a second set of linear equations, block 508. Once defined, the second set of linear equations are solved to find the solution to both the second set, and the first set, of quadratic equations, block 510.

As appreciated by those skilled in the art, the operations and methodologies illustrated in FIGS. 1, 2 and 5 may be performed sequentially, or in parallel, using one or more processors to determine specific values. Further, the methodologies disclosed herein may be applied to analyze third order systems as well. For example, in one embodiment, third order equations are used when solving for State Estimation (SE) using the Least Squares technique.

As can be appreciated by referring once again to FIG. 4, processing system 406 may be in two-way communication with each of sources 402 to provide results of the processing to sources 402. Processor system 406 may further receive or transmit data over one or more network connections from a server or servers, over one or more global computer communications networks such as: the Internet, an Intranet, a wide area network (WAN), a metropolitan area network (MAN), a local area network (LAN), a terrestrial broadcast system, a cable network, a satellite network, a wireless network, microwave, or a telephone network (POTS), as well as portions or combinations of these and other types of networks. As will be appreciated, networks 404 and 414 may also be internal networks, e.g., ISA bus, micro channel bus, PCI bus, PCMCIA bus, etc. Alternatively, the networks 404, 414 may also be one or more internal connections of a circuit, a circuit card or other device, as well as portions and combinations of these and other communication media or an external network, e.g., the Internet and Intranet.

Changes may be made in the above methods, devices and structures without departing from the scope hereof. It should thus be noted that the matter contained in the above description and/or shown in the accompanying drawings should be interpreted as illustrative and not in a limiting sense. The following claims are intended to cover all generic and specific features described herein, as well as all statements of the scope of the present method, device and structure, which, as a matter of language, might be said to fall therebetween. 

1. A method for analyzing a power system, comprising: formulating a set of non-linear equations representative of a power system and having parameters and unknowns; transforming the set of non-linear equations into a corresponding first set of quadratic equations; changing variables of the first set of quadratic equations to define a second set of quadratic equations; and solving the second set of quadratic equations to define a solution set for the power system unknowns.
 2. The method of claim 1, further comprising: determining when the solution set is outside a predetermined tolerance; and iteratively calculating the solution set to be within the predetermined tolerance.
 3. The method of claim 1, wherein the power system comprises one of: AC power flow, AC state estimation, optimal power flow, stability analysis, economic dispatch, and unit commitment.
 4. A system for analyzing a power system comprising: an algorithm for transforming a set of non-linear equations, representative of a power system and having parameters and unknowns, into a first set of linear equations having a coefficients matrix, a variables vector and a right hand side vector; an algorithm for solving for a first variable set in terms of a second variable set in the first set of linear equations, for formulating a second set of quadratic equations, and for transforming the second set of quadratic equations into a second set of linear equations wherein a third variable set is defined; an algorithm for calculating the first, second and third variable sets, and for calculating a solution set for the set of non-linear equations; and a processor for executing the transforming, the solving and the calculating algorithms, and for and applying the solution set to the set of non-linear equations to solve for the unknowns.
 5. The system of claim 4, the power system being selected from a group consisting of: AC power flow, AC state estimation, optimal power flow, stability analysis, economic dispatch, and unit commitment.
 6. The system of claim 4, further comprising: an algorithm for determining when the calculated solution set is outside a predetermined tolerance; and an algorithm for iteratively recalculating the solution set to be within the predetermined tolerance.
 7. A method for analyzing a power system comprising: formulating a set of nonlinear equations representative of a power system and having parameters and unknowns; transforming the set of non-linear equations into a corresponding first set of quadratic equations; expressing the first set of quadratic equations as a first set of linear equations having a coefficients matrix, a variables vector and a right hand side vector; defining a first variable set in the first set of linear equations in terms of a second variable set; formulating a second set of quadratic equations in terms of the second variable set; converting the second set of quadratic equations into a second set of linear equations defined in terms of a third variable set; solving for the third variable set; and calculating the first variable set and the second variable set to define a solution set for the power system unknowns.
 8. The method of claim 7, wherein the power system comprises one of: AC power flow, AC state estimation, optimal power flow, stability analysis, economic dispatch, and unit commitment.
 9. The method of claim 7, further comprising: determining when the solution set is outside a predetermined tolerance; and iteratively recalculating the solution set to be within the predetermined tolerance.
 10. A method for analyzing a power system, comprising: defining a set of non-linear equations representative of a power system and containing power system parameters and unknowns; inputting power system parameters and inputting power system measurements; selecting between an iterative and a non-iterative analytical approach to calculate a solution set for the set of non-linear equations, wherein the non-iterative analytical approach is selected if an adequate set of synchronous power system measurements are available; otherwise estimating the required number of power system unknowns; calculating the solution set for the set of non-linear equations, to solve for the remaining power system unknowns; and iterating the calculation of the solution set, when the iterative analytical approach has been selected, to converge on a solution set within a predetermined tolerance.
 11. The method of claim 10, wherein the power system formulation or representation is selected from a group consisting of: AC power flow, AC state estimation, optimal power flow, stability analysis, economic dispatch, and unit commitment.
 12. The method of claim 10, wherein the step of inputting comprises directly measuring the power system with an adequate number of its variables.
 13. The method of claim 10, wherein the step of inputting comprises inputting one or more of synchronous phasors, asynchronous supervisory control and data acquisition measurements.
 14. A software product comprising instructions, stored on computer-readable media, wherein the instructions, when executed by a computer, perform steps for power systems analysis, comprising: instructions for formulating a set of non-linear equations representative of a power system and containing power system parameters and unknowns; instructions for transforming the set of non-linear equations into a corresponding first set of quadratic equations; instructions for subsequently changing variables of the first set of quadratic equations to define a second set of quadratic equations; and instructions for solving the second set of quadratic equations to define a solution set for the power system unknowns.
 15. The software product of claim 14, the power system being selected from a group consisting of: AC power flow, AC state estimation, optimal power flow, stability analysis, economic dispatch, and unit commitment.
 16. The software product of claim 14, further comprising: instructions for determining when the defined solution set is outside a predetermined tolerance; and instructions for iteratively redefining the solution set to be within the predetermined tolerance.
 17. The software product of claim 14, further comprising instructions for directly measuring the power system with an adequate number of its variables. 